Preference relation

In microeconomics, a preference relation is generally referred to as a ranking in which two bundles of goods (“alternatives”) are arranged according to how they are preferred by an individual or a group of individuals. Formally, a preference relation is a binary relation . For example, there is a preference relation (the so-called preference-indifference relation, also: weak preference relation), which indicates that its first component is perceived as strictly better than or as good as the second. For example, if a person prefers an alternative (weak) to , then the tuple is included in the set (the index is intended to indicate that these are the preferences of the person ). ${\displaystyle R}$ ${\displaystyle i}$ ${\displaystyle \mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}}$ ${\displaystyle (\mathbf {x} _{a},\mathbf {x} _{b})}$ ${\displaystyle R_{i}}$ ${\displaystyle i}$ ${\displaystyle i}$

Other preference relations are the strict preference relation ("strictly better than") and the indifference relation ("as good as"); A separate definition of the opposite constellation (“worse than or equally good as” or “strictly worse than”) is usually not given, since the underlying preference structures can also be formulated in the manner defined here by interchanging the components. ${\displaystyle P}$ ${\displaystyle I}$

A preference relation is called a preference order if it meets certain minimum requirements (see section # preference order ). If this is the case, one speaks of the rationality of the preference-indifference relation. The order of preferences is an important concept of rationality within economics. Another system of axioms for a rational decision maker comes from John von Neumann and Oskar Morgenstern .

history

The first axiomatic foundation of the preference relation was presented by Ragnar Frisch in 1926 . After the pioneering work by Frisch in the 1920s, the main focus was on how to map a preference structure to a real-valued function . This was achieved through the concept of utility function , a mathematical modeling of preferences. In 1954, Gérard Debreu made an important contribution to the connection between the preference relation and utility function: his representation theorem (also called Debreu's theorem).

definition

One starts with a set in which all existing bundles of goods ("alternatives") are included: ${\displaystyle X}$ ${\displaystyle n}$

${\displaystyle X=\{\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}\}}$

The are bundle of goods and thus -tuples , so that, for example, indicating the quantity of material 5 in the bundle of goods. 3 For tuples, unlike sets , the order of the objects plays a role. ${\displaystyle \mathbf {x} _{i}}$ ${\displaystyle m}$ ${\displaystyle (x_{i}^{1},\dotsc ,x_{i}^{m})}$ ${\displaystyle x_{3}^{5}}$

Preference relations are binary relations , that is, they are subsets of . In the following, only the so-called preference-indifference relation ( as described) will be considered. In this definition , all ordered pairs are included, for which it applies that weak is preferred over. From now on the notation for . You can also read straight away as “is better than or as good as”. ${\displaystyle X}$ ${\displaystyle X\times X=\{(\mathbf {x} _{1},\mathbf {x} _{1}),(\mathbf {x} _{1},\mathbf {x} _{2}),\dotsc ,(\mathbf {x} _{2},\mathbf {x} _{1}),(\mathbf {x} _{2},\mathbf {x} _{2}),\dotsc \}}$ ${\displaystyle R}$ ${\displaystyle R\subseteq X\times X}$ ${\displaystyle R}$ ${\displaystyle (\mathbf {x} _{a},\mathbf {x} _{b})}$ ${\displaystyle \mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{b}}$ ${\displaystyle (\mathbf {x} _{a},\mathbf {x} _{b})\in R}$ ${\displaystyle R}$

By again subsets of - two other relations are - induced. On the one hand the indifference relation , on the other hand the preference relation . Their meaning results from that of : For two alternatives and is exactly then or , if and at the same time . can then be read as “is as good as”. The same applies to : For two alternatives and is exactly then or if , but not at the same time . one reads as "is better than". ${\displaystyle R}$ ${\displaystyle X\times X}$ ${\displaystyle I}$ ${\displaystyle P}$ ${\displaystyle R}$ ${\displaystyle \mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}}$ ${\displaystyle (\mathbf {x} _{a},\mathbf {x} _{b})\in I}$ ${\displaystyle \mathbf {x} _{a}I\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{a}}$ ${\displaystyle I}$ ${\displaystyle P}$ ${\displaystyle \mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}}$ ${\displaystyle (\mathbf {x} _{a},\mathbf {x} _{b})\in P}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{a}}$ ${\displaystyle P}$

Instead of letters , and for the relations, symbols are also used. It is then as well as and . ${\displaystyle R}$ ${\displaystyle P}$ ${\displaystyle I}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{b}\Leftrightarrow \mathbf {x} _{a}\succsim \mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{b}\Leftrightarrow \mathbf {x} _{a}\succ \mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{a}I\mathbf {x} _{b}\Leftrightarrow \mathbf {x} _{a}\sim \mathbf {x} _{b}}$

If you want to express that you are referring to the preference structure of a specific person , you can index the relation accordingly; for example, this means that the person strictly prefers the alternative . ${\displaystyle i}$ ${\displaystyle x_{a}P_{i}x_{b}}$ ${\displaystyle i}$ ${\displaystyle x_{a}}$ ${\displaystyle x_{b}}$

properties

Depending on its individual nature, the preference-indifference relation can be examined for the following properties, for example: ${\displaystyle R}$

• Completeness: or (or both) ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b}\in X:\mathbf {x} _{a}R\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{a}}$

This ensures that there is actually a ranking for each alternative; However, the completeness property does not mean that there actually has to be a strict preference - rather, two alternatives can be perceived as being equivalent without contradicting this condition.

• Transitivity : ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b},\mathbf {x} _{c}\in X:\mathbf {x} _{a}R\mathbf {x} _{b}\wedge \mathbf {x} _{b}R\mathbf {x} _{c}\Rightarrow \mathbf {x} _{a}R\mathbf {x} _{c}}$
  

  Circular conclusion (violation of the transitivity assumption), which was caused by intersecting "indifference curves"

The property prevents so-called circular preferences from occurring. The following example shows that the preference structure would be circular without its validity: Think of a person who is at least as fond of apples as pears and pears as fond of lemons. If she didn't like apples at least as much as lemons, as required by the transitivity property, then logically the opposite would have to apply: She would rather have lemons than apples. It was assumed, however, that she is at least as fond of pears as lemons, etc., so that the search for the preferred option would be fruitless without the assumption of transitivity.

• Reflexivity : ${\displaystyle \forall \mathbf {x} _{a}\in X:\mathbf {x} _{a}R\mathbf {x} _{a}}$

What is meant is that an alternative is always rated the same regardless of the situation (in short, it applies ). The property of reflexivity is classically mentioned in the same series as the two preceding axioms, but it actually follows from the property of completeness. ${\displaystyle \mathbf {x} _{a}I\mathbf {x} _{a}}$

• Continuity : Thefollowing applies toall: The quantities(upper contourquantity) and(lower contour quantity) are complete with regard to ${\displaystyle \mathbf {x} _{b}\in X}$ ${\displaystyle \{\mathbf {x} _{a}\in X|\mathbf {x} _{a}R\mathbf {x} _{b}\}}$ ${\displaystyle \{\mathbf {x} _{a}\in X|\mathbf {x} _{b}R\mathbf {x} _{a}\}}$ ${\displaystyle X.}$
• (Weak) monotony : ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b}\in X:(\mathbf {x} _{a}\geq \mathbf {x} _{b})\Rightarrow \mathbf {x} _{a}R\mathbf {x} _{b}}$

To understand it, you have to recall that the alternatives represent bundles of goods: If a bundle of goods contains at least as large a number of every good as a bundle of goods , then it is rated at least as good as is defined analogously ${\displaystyle \mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}.}$

• Strict monotony: ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b}\in X:(\mathbf {x} _{a}\geq \mathbf {x} _{b})\wedge (\mathbf {x} _{a}\neq \mathbf {x} _{b})\Rightarrow \mathbf {x} _{a}P\mathbf {x} _{b}}$

If a bundle of goods of every good contains at least as large a number as a bundle of goods of at least one good, but even a strictly larger number, then it is strictly preferred. Every strictly monotonic preference relation is therefore also (weakly) monotonic. ${\displaystyle \mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b},}$ ${\displaystyle \mathbf {x} _{b}}$

• Unsaturation : For everyone there is one with the property In other words, for every choice there is a better alternative. ${\displaystyle \mathbf {x} _{a}\in X}$ ${\displaystyle \mathbf {x} _{b}\in X}$ ${\displaystyle \mathbf {x} _{b}P\mathbf {x} _{a}.}$
• Local unsaturation: For each and every environment around there is one with the property Formal: Compared to unsaturation, local unsaturation is a stricter assumption because there must be a better alternative in every area around the starting point, no matter how small. On the other hand, local unsaturation is a weaker assumption than strict monotony because it allows the existence of bads, i.e. goods that reduce the benefit of the individual. ${\displaystyle \mathbf {x} _{a}\in X}$ ${\displaystyle \mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}P\mathbf {x} _{a}.}$ ${\displaystyle \forall \mathbf {x} _{a}\in X\;\forall \epsilon >0\;\exists \mathbf {x} _{b}\in X:\left(\left\Vert \mathbf {x} _{a}-\mathbf {x} _{b}\right\Vert \leq \epsilon \right)\wedge \mathbf {x} _{b}P\mathbf {x} _{a}}$
• Convexity : The convex combination of two bundles of goods rated as equally good is rated at least as good or better than one of the bundles of goods. When viewed graphically, the associated indifference curves are strictly convex or at least linear. The upper contour sets are convex sets for all. ${\displaystyle \forall \mathbf {x} ,\mathbf {y} \in X:\mathbf {x} I\mathbf {y} \Rightarrow \forall t\in (0,1):[t\mathbf {x} +(1-t)\mathbf {y} ]R\mathbf {x} .}$ ${\displaystyle \{\mathbf {x} _{b}\in X|\mathbf {x} _{b}R\mathbf {x} _{a}\}}$ ${\displaystyle \mathbf {x} _{a}\in X}$
• Strict convexity: A convex combination of two not identical but equally well rated bundles is preferred to the one bundle. In the graphical view, the associated indifference curves are strictly convex. ${\displaystyle \forall \mathbf {x} ,\mathbf {y} (\mathbf {x} \neq \mathbf {y} )\in X:\mathbf {x} I\mathbf {y} \Rightarrow \forall t\in (0,1):[t\mathbf {x} +(1-t)\mathbf {y} ]P\mathbf {x} .}$
• Homothety : Geometrically, this property implies that the slopes of the indifference curves remain constant along an original ray. Therefore, the income-consumption curves are linear: As income increases, all goods are in demand in unchanged proportions. ${\displaystyle \forall \mathbf {x} ,\mathbf {y} \in X,t>0:\mathbf {x} I\mathbf {y} \Rightarrow (t\mathbf {x} )I(t\mathbf {y} ).}$

Order of preference

The first two properties are particularly important. With them the following applies:

Rationality of the preference-indifference relation: A preference-indifference relation R is rational if and only if it is complete and transitive. They are then also referred to as the order of preference.

It can be shown that the rationality of R also has important effects on the relations it induces:

Implications for the preference and indifference relation : If the preference-indifference relation R is rational, then the following applies to the relations I and P induced by it :

1. P is
   • transitive: and ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b},\mathbf {x} _{c}:\mathbf {x} _{a}P\mathbf {x} _{b}\wedge \mathbf {x} _{b}P\mathbf {x} _{c}\Rightarrow \mathbf {x} _{a}P\mathbf {x} _{c}}$
   • asymmetrically: . ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b}:\mathbf {x} _{a}P\mathbf {x} _{b}\Rightarrow \neg \mathbf {x} _{b}P\mathbf {x} _{a}}$
2. I is
   • reflexive ; ${\displaystyle \forall \mathbf {x} _{a}:\mathbf {x} _{a}R\mathbf {x} _{a}}$
   • transitive: and ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b},\mathbf {x} _{c}:\mathbf {x} _{a}I\mathbf {x} _{b}\wedge \mathbf {x} _{b}I\mathbf {x} _{c}\Rightarrow \mathbf {x} _{a}I\mathbf {x} _{c}}$
   • symmetrical: . ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b}:\mathbf {x} _{a}I\mathbf {x} _{b}\Rightarrow \mathbf {x} _{b}I\mathbf {x} _{a}}$

In support of the see section #Implikationen of R and P I .

Implications for the utility function

Mathematically, it is often easier to represent orders of preference using utility functions. A function is called a utility function , which represents the order of preference if . The question arises whether every order of preference can be represented by a utility function. This is not the case, but the following assumptions are sufficient: ${\displaystyle u}$ ${\displaystyle R}$ ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b}\in X:\mathbf {x} _{a}R\mathbf {x} _{b}\Leftrightarrow u(\mathbf {x} _{a})\geq u(\mathbf {x} _{b})}$

Debreus representation theorem: If is a connected subset of , then every continuous order of preference can be represented by a continuous real-valued utility function . ${\displaystyle X}$ ${\displaystyle R^{n}}$ ${\displaystyle X}$ ${\displaystyle u}$

Alternative assumptions are derived in the literature that ensure the existence of a utility function. The above assumptions are therefore sufficient but not necessary. If the set of alternatives contains a finite or countably infinite number of elements, a preference order defined on it can always be represented without additional assumptions.

Example : If with two alternatives {a, b} the first is strictly preferred over the second, this order of preference can be represented by the utility function u (a) = 2 and u (b) = 1.

Lexicographical order of preferences

→ See also: Lexicographical order
Consider two bundles of goods and as an example . The order of preference for these bundles of goods is called lexicographically if the following applies: ${\displaystyle \mathbf {x} =(x_{1},x_{2})}$ ${\displaystyle \mathbf {y} =(y_{1},y_{2})}$

1. From follows ${\displaystyle x^{1}>y^{1}}$ ${\displaystyle \mathbf {x} P\mathbf {y} ,}$
2. out and follows ${\displaystyle x^{1}=y^{1}}$ ${\displaystyle x^{2}>y^{2}}$ ${\displaystyle \mathbf {x} P\mathbf {y} .}$

If a bundle of goods contains more of good 1, then it is strictly preferred, regardless of the quantity of good 2 it contains. Only if both bundles of goods contain good 1 in the same amount does the amount of good 2 matter. From the point of view of the considered individual, good 1 is much more important than good 2.

Lexicographical preference orders are not continuous and therefore cannot be represented by utility functions. That they are not continuous can be seen as follows: Let be a sequence of bundles of goods, the members of which satisfy the conditions and . This implies for all the sequence converges to a limit value with and then what violates the assumption of closure or continuity does not apply . ${\displaystyle \mathbf {x} _{n}}$ ${\displaystyle x_{n}^{1}>y^{1}}$ ${\displaystyle x_{n}^{2}<y^{2}}$ ${\displaystyle \mathbf {x} _{n}R\mathbf {y} }$ ${\displaystyle n.}$ ${\displaystyle x_{1}^{\infty }=y_{1}}$ ${\displaystyle x_{2}^{\infty }<y_{2},}$ ${\displaystyle \mathbf {x} ^{\infty }R\mathbf {y} ,}$

Mathematical basics; formal supplements

Definitions used in the following (partial repetition of the above) for a general non-empty set X (B a binary relation on X):

• Completeness: or (or both) ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b}\in X:\mathbf {x} _{a}B\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}B\mathbf {x} _{a}}$
• Reflexivity: ${\displaystyle \forall \mathbf {x} _{a}\in X:\mathbf {x} _{a}B\mathbf {x} _{a}}$
• Irreflectivity: ${\displaystyle \forall \mathbf {x} _{a}\in X:\neg \mathbf {x} _{a}B\mathbf {x} _{a}}$
• Symmetry: ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b}\in X:\mathbf {x} _{a}B\mathbf {x} _{b}\Rightarrow \mathbf {x} _{b}B\mathbf {x} _{a}}$
• Asymmetry: ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b}\in X:\mathbf {x} _{a}B\mathbf {x} _{b}\Rightarrow \neg \mathbf {x} _{b}B\mathbf {x} _{a}}$
• Transitivity: ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b},\mathbf {x} _{c}\in X:\mathbf {x} _{a}B\mathbf {x} _{b}\wedge \mathbf {x} _{b}B\mathbf {x} _{c}\Rightarrow \mathbf {x} _{a}B\mathbf {x} _{c}}$
• Negative transitivity : ${\displaystyle \forall \mathbf {x} _{a},\mathbf {x} _{b},\mathbf {x} _{c}\in X:\left(\mathbf {x} _{a}B\mathbf {x} _{c}\Rightarrow \mathbf {x} _{a}B\mathbf {x} _{b}\vee \mathbf {x} _{b}B\mathbf {x} _{c}\right)}$

Implications of R for P and I.

The outlined concept can be generalized so that, among other things, it is possible to put the three relations considered here in a formal context.

We agree that

• the preference-indifference relation is a complete quasi-order, that is, it is complete, reflexive and transitive (economically this corresponds to our definition of the preference order). ${\displaystyle R}$
• ${\displaystyle P}$is the asymmetrical part of the quasi-order , that is, the following applies: ${\displaystyle R}$

${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{b}\Leftrightarrow \left(\mathbf {x} _{a}R\mathbf {x} _{b}\wedge \neg \mathbf {x} _{b}R\mathbf {x} _{a}\right)}$

• ${\displaystyle I}$is the symmetrical part of the quasi-order , which means that: ${\displaystyle R}$

${\displaystyle \mathbf {x} _{a}I\mathbf {x} _{b}\Leftrightarrow \left(\mathbf {x} _{a}R\mathbf {x} _{b}\wedge \mathbf {x} _{b}R\mathbf {x} _{a}\right)}$

The following central statements apply:

Implications of the property as a quasi-order: Let a complete quasi-order on a non-empty one with an asymmetrical part and a symmetrical part . Then: ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle P}$ ${\displaystyle I}$

1. ${\displaystyle P}$ is (a) irreflexive, (b) asymmetric, (c) negative transitive, and (d) transitive
2. ${\displaystyle I}$is an equivalence relation , i.e. I is (a) symmetric, (b) reflexive and (c) transitive.
3. ${\displaystyle \neg \mathbf {x} _{a}P\mathbf {x} _{b}\Leftrightarrow \mathbf {x} _{b}R\mathbf {x} _{a}}$or equivalent , that is, is the negation of (and vice versa). ${\displaystyle \neg \mathbf {x} _{b}R\mathbf {x} _{a}\Leftrightarrow \mathbf {x} _{a}P\mathbf {x} _{b}}$ ${\displaystyle R}$ ${\displaystyle P}$
4. Exactly one of the following statements is true: (a) , (b) , (c) . ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}P\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{a}I\mathbf {x} _{b}}$

Evidence to 1. and 2 .: (1a) Be and any item from and be . Then according to the definition of P, that and at the same time , a contradiction, that is . (1b) The asymmetry already results from the definition of the asymmetrical part. (1c) Be such that ; is to be shown in the following that then also . From the assumptions and the definition of P it follows that as well as . Since and , because of the transitivity of R, also holds - consequently it is now sufficient to show that , to prove that not at the same time . Proof by contradiction: Would , then also , because according to the above conclusion from the definition of P, and thus also according to the transitivity property of R. But this contradicts the above insight that , is consequent and together with (see above) it indeed follows what was to be shown. (1d) Let . It has to be shown that either or . Well , then too (asymmetry). Together with the original assumption, however, it then follows: (transitivity), which was to be proven. (2a, b) Follows directly from the definition of the symmetrical part. (2c) Be such that ; is to be shown in the following that then also . From the assumptions and the definition of I it follows that as well as . According to the transitivity of R, it follows that also (left sides) and (right sides). However, according to the definition of the symmetrical part, this also applies to what was to be shown. ${\displaystyle \mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}}$ ${\displaystyle X}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{a}}$ ${\displaystyle \neg \mathbf {x} _{a}R\mathbf {x} _{a}}$ ${\displaystyle \neg \mathbf {x} _{a}P\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{a},\mathbf {x} _{b},\mathbf {x} _{c}\in X}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{b}\wedge \mathbf {x} _{b}P\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{b}\wedge \neg \mathbf {x} _{b}R\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{c}\wedge \neg \mathbf {x} _{c}R\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{c}R\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{c}R\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{a}}$ ${\displaystyle \neg \mathbf {x} _{b}R\mathbf {x} _{a}}$ ${\displaystyle \neg \mathbf {x} _{c}R\mathbf {x} _{a}}$ ${\displaystyle \neg \mathbf {x} _{b}R\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}P\mathbf {x} _{c}}$ ${\displaystyle \neg \mathbf {x} _{a}P\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}P\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}P\mathbf {x} _{a}\wedge \mathbf {x} _{a}P\mathbf {x} _{c}\Rightarrow \mathbf {x} _{b}P\mathbf {x} _{c}}$
${\displaystyle \mathbf {x} _{a},\mathbf {x} _{b},\mathbf {x} _{c}\in X}$ ${\displaystyle \mathbf {x} _{a}I\mathbf {x} _{b}\wedge \mathbf {x} _{b}I\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{a}I\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{b}\wedge \mathbf {x} _{b}R\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{c}\wedge \mathbf {x} _{c}R\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{c}}$ ${\displaystyle \mathbf {x} _{c}R\mathbf {x} _{a}}$ ${\displaystyle \mathbf {x} _{a}I\mathbf {x} _{c}}$

Implications of P for R and I.

The third statement (3) of the Theoremkastens in the preceding section ( "implication of the property as a quasi-order") can be a significant relationship between P and R may be revealed. In the economic context, the preference-indifference relation is precisely the negation of a strict preference relation. However, this suggests that the preferences can also be determined differently than before on the basis of the strict preference relation and not only, as described in the rest of this article, on the basis of the preference-indifference relation. In practice, one could ask whether it should not be possible instead of “asking” consumers about their weak preferences (“Do you like ice cream at least as much as cake?” And “Do you like cake at least as much as ice cream?”), But to begin with the strict preferences (“Do you prefer ice cream to cake?”) and derive from this, among other things, the preference-indifference relation. However, this is not always a quasi-order (or a preference order), as the following example illustrates.

Example: Let and , . Define with . Let then the induced preference-indifference relation R is given by . It is not transitive, as can be verified by choosing suitable examples. Consider , for example, the following combination of goods:, and , then is and , but . ${\displaystyle X=\mathbb {R} _{+}}$ ${\displaystyle g:X\rightarrow \mathbb {R} }$ ${\displaystyle \mathbf {x} \mapsto g(\mathbf {x} )}$ ${\displaystyle \mathbf {x} _{a}P\mathbf {x} _{b}\Leftrightarrow g(\mathbf {x} _{a})+c>g(\mathbf {x} _{b})}$ ${\displaystyle c>0}$ ${\displaystyle g(\mathbf {x} )=\mathbf {x} }$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{b}\Leftrightarrow \mathbf {x} _{b}-1\leq \mathbf {x} _{a}}$ ${\displaystyle c=1}$ ${\displaystyle \mathbf {x} _{a}=0}$ ${\displaystyle \mathbf {x} _{b}=1}$ ${\displaystyle \mathbf {x} _{c}=2}$ ${\displaystyle \mathbf {x} _{a}R\mathbf {x} _{b}}$ ${\displaystyle \mathbf {x} _{b}R\mathbf {x} _{c}}$ ${\displaystyle \neg \mathbf {x} _{a}R\mathbf {x} _{c}}$

The following theorem forms the starting point for initially technical considerations in the first part; the last statement describes a concrete criterion under which a given strict preference relation induces a preference-indifference relation that fulfills the rationality criterion (i.e. is a preference order).

Implications of the property as a strict order of preference: Let be a binary relation on non-empty and let be the negation of . Then: ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle P}$

1. ${\displaystyle R}$is transitive if and only if negative is transitive. ${\displaystyle P}$
2. If, in addition, asymmetric and negative transitive is on , then a quasi-order is on and is its asymmetric part. ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle P}$

See also

• Arrow theorem
• Pareto optimum

literature

• Ragnar Frisch: Sur unproblemème d'économie pure. In: Norsk Matematisk Forenings Skrifter. Oslo 1 (16), 1926, pp. 1-40.
• Gerard Debreu: Representation of a preference ordering by a numerical function. Decision processes 3, 1954, pp. 159-165.
• Fuad Aleskerov, Denis Bouyssou, Bernard Monjardet: Utility Maximization, Choice and Preference. 2nd Edition. Springer, Heidelberg et al. 2007, ISBN 978-3-540-34182-6 .
• Anton Barten, Volker Böhm: Consumer Theory. In: Kenneth J. Arrow, Michael D. Intrilligator (Eds.): Handbook of Mathematical Economics. Volume 2, North Holland, Amsterdam 1982, ISBN 0-444-86127-0 , pp. 382-429.
• Friedrich Breyer: Microeconomics. An introduction. 3. Edition. Springer, Heidelberg et al. 2007, ISBN 978-3-540-69230-0 .
• Geoffrey A. Jehle, Philip J. Reny: Advanced Microeconomic Theory. 3. Edition. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Andreu Mas-Colell, Michael Whinston, Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-19-507340-1 .
• James C. Moore: Mathematical methods for economic theory. Volume 1, Springer, Berlin et al. 1995, ISBN 3-540-66235-9 .
• James C. Moore: General equilibrium and welfare economics. An introduction. Springer, Berlin et al. 2007, ISBN 978-3-540-31407-3 (also online: doi: 10.1007 / 978-3-540-32223-8 ).
• Hal Varian : Microeconomic Analysis. WW Norton, New York / London 1992, ISBN 0-393-95735-7 .

Individual evidence

1. ^ Axioms of rational decision-making , definition in the Gabler Wirtschaftslexikon.
2. ↑ Wolfgang J. Fellner: From the goods economy to the activity economy: time use and endogenous preferences in a consumption model . Springer-Verlag, 2014, p. 10.
3. ^ Ragnar Frisch: Sur unproblemème d'économie pure. In: Norsk Matematisk Forenings Skrifter. Oslo 1 (16), 1926, pp. 1-40.
4. ^ Gerard Debreu: Representation of a preference ordering by a numerical function. Decision processes 3, 1954, pp. 159-165.
5. ↑ Largely based on Jehle / Reny 2011, pp. 4–12.
6. ↑ ^{a b} The definitions of monotony follow Varian 1992 (p. 96), but are defined differently in the literature. Barten / Böhm 1982 (p. 390 f.) Do not even introduce the concept of weak monotony in the sense presented here, but define the property “monotony” according to the local definition of strict monotony. Mas-Colell / Whinston / Green 1995 (p. 42) use the same definition as here for the strict monotonicity property and define for the (weak) monotony that R is (weakly) monotonic if ${\displaystyle X,}$ ${\displaystyle (\mathbf {x} _{b}\in X)\wedge (\mathbf {x} _{a}>\mathbf {x} _{b})\Rightarrow \mathbf {x} _{b}P\mathbf {x} _{a}.}$
7. ↑ See Barten / Böhm 1982, p. 391 and Mas-Colell / Whinston / Green 1995, p. 44.
8. ↑ See Barten / Böhm 1982; Breyer 2007, p. 117; for the concept of the “rational order of preference” see Mas-Colell / Whinston / Green 1995, p. 6. Other authors link the entire term “preference-indifference relation” (or “weak preference relation”) to the fulfillment of this Condition. See for example Jehle / Reny 2011, p. 6.
9. ↑ See Mas-Colell / Whinston / Green 1995, p. 7.
10. ^ G. Debreu: Theory of Value . Yale University Press 1959, p. 59.
11. ^ G. Herden: On the Existence of Utility Functions. In: Mathematical Social Sciences. 17, 1989, pp. 297-313.
12. ↑ On this in detail Mas-Colell / Whinston / Green 1995, p. 46 f.
13. ↑ See Moore 1995, pp. 21, 23 ff.
14. ↑ See Moore 2007, p. 6 f.
15. ↑ See Moore 2007, p. 7; Aleskerov / Bouyssou / Monjardet 2007, p. 24.
16. ^ Similar to Moore 1995, p. 23.
17. ↑ See Moore 2007, p. 8 ff.